An extended class of minimax generalized Bayes estimators of regression coefficients

We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman [Y. Maruyama, W.E. Strawderman, A new class of generalized Bayes minimax ridge regression estimators, Ann. Statist., 33 (2005) 1753–1770] to include non-monotone shrinkage functions.     

Minimaxity and admissibility of the product limit estimator

In this article we examine the minimaxity and admissibility of the product limit (PL) estimator under the loss function% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbGaaiikai% aadAeacaGGSaGabmOrayaajaGaaiykaiabg2da9maapeaabaGaaiik% aiaadAeacaGGOaGaamiDaiaacMcaaSqabeqaniabgUIiYdGccqGHsi% slceWGgbGbaKaacaGGOaGaamiDaiaacMcacaGGPaWaaWbaaSqabeaa% caaIYaaaaOGaamOramaaCaaaleqabaaccmGae8xSdegaaOGaaiikai% aadshacaGGPaGaaiikaiaaigdacqGHsislcaWGgbGaaiikaiaadsha% caGGPaGaaiykamaaCaaaleqabaGaeqOSdigaaOGaamizaiaadEfaca% GGOaGaamiDaiaacMcaaaa!5992!\[L(F,\hat F) = \int {(F(t)} - \hat F(t))^2 F^\alpha (t)(1 - F(t))^\beta dW(t)\].To avoid some pathological and uninteresting cases, we restrict the parameter space to ={F: F(ymin) }, where (0, 1) and y ₁,...y,_n are the censoring times. Under this set up, we obtain several interesting results. When y ₁=···=y _n, we prove the following results: the PL estimator is admissible under the above loss function for , {–1, 0}; if n=1, ==–1, the PL estimator is minimax iff dW ({y})=0; and if n2, , {–1, 0}, the PL estimator is not minimax for certain ranges of . For the general case of a random right censorship model it is shown that the PL estimator is neither admissible nor minimax. Some additional results are also indicated.Partially supported by the Governor's Challenge Grant.Part of the work was done while the author was visiting William Paterson College.     

Estimation of covariance matrices in fixed and mixed effects linear models

The estimation of the covariance matrix or the multivariate components of variance is considered in the multivariate linear regression models with effects being fixed or random. In this paper, we propose a new method to show that usual unbiased estimators are improved on by the truncated estimators. The method is based on the Stein–Haff identity, namely the integration by parts in the Wishart distribution, and it allows us to handle the general types of scale-equivariant estimators as well as the general fixed or mixed effects linear models.     

Asymptotic Results on a General Class of Empirical Statistics: Power and Confidence Interval Properties

We consider a very general class of empirical statistics that includes (a) empirical discrepancy (ED) statistics, (b) generalized empirical exponential family likelihood statistics, (c) generalized empirical likelihood statistics, (d) empirical statistics arising from Bayesian considerations, and (e) Bartlett-type adjusted versions of ED statistics. With reference to this general class, we investigate higher order asymptotics on power and expected lengths of confidence intervals. For (b)-(e), such results have been hitherto unexplored. Furthermore, our findings help in understanding the presently known results on the subclass (a) from a wider perspective.     

A note on shrinkage wavelet estimation in Bayesian analysis

In this short note the closed form of the soft wavelet shrinkage estimator is derived, extending the work of Huang (2002) for the scale mixture of normal distributions.     

Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models. The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given.     

Minimax invariant estimator of a continuous distribution function

Consider the problems of the continuous invariant estimation of a distribution function with a wide class of loss functions. It has been conjectured for long that the best invariant estimator is minimax for all sample sizes n1. This conjecture is proved in this short note.Partially supported by National Science Foundation Grant DMS 9001194.